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Properties

Label 2-825-1.1-c5-0-138
Degree 22
Conductor 825825
Sign −1-1
Analytic cond. 132.316132.316
Root an. cond. 11.502811.5028
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.4·2-s + 9·3-s + 77.9·4-s − 94.3·6-s + 152.·7-s − 481.·8-s + 81·9-s + 121·11-s + 701.·12-s + 260.·13-s − 1.60e3·14-s + 2.55e3·16-s − 586.·17-s − 849.·18-s − 614.·19-s + 1.37e3·21-s − 1.26e3·22-s + 1.37e3·23-s − 4.33e3·24-s − 2.72e3·26-s + 729·27-s + 1.19e4·28-s − 4.06e3·29-s − 2.68e3·31-s − 1.13e4·32-s + 1.08e3·33-s + 6.15e3·34-s + ⋯
L(s)  = 1  − 1.85·2-s + 0.577·3-s + 2.43·4-s − 1.07·6-s + 1.17·7-s − 2.66·8-s + 0.333·9-s + 0.301·11-s + 1.40·12-s + 0.426·13-s − 2.18·14-s + 2.49·16-s − 0.492·17-s − 0.617·18-s − 0.390·19-s + 0.681·21-s − 0.558·22-s + 0.542·23-s − 1.53·24-s − 0.791·26-s + 0.192·27-s + 2.87·28-s − 0.896·29-s − 0.502·31-s − 1.96·32-s + 0.174·33-s + 0.912·34-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: −1-1
Analytic conductor: 132.316132.316
Root analytic conductor: 11.502811.5028
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 825, ( :5/2), −1)(2,\ 825,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1−9T 1 - 9T
5 1 1
11 1−121T 1 - 121T
good2 1+10.4T+32T2 1 + 10.4T + 32T^{2}
7 1−152.T+1.68e4T2 1 - 152.T + 1.68e4T^{2}
13 1−260.T+3.71e5T2 1 - 260.T + 3.71e5T^{2}
17 1+586.T+1.41e6T2 1 + 586.T + 1.41e6T^{2}
19 1+614.T+2.47e6T2 1 + 614.T + 2.47e6T^{2}
23 1−1.37e3T+6.43e6T2 1 - 1.37e3T + 6.43e6T^{2}
29 1+4.06e3T+2.05e7T2 1 + 4.06e3T + 2.05e7T^{2}
31 1+2.68e3T+2.86e7T2 1 + 2.68e3T + 2.86e7T^{2}
37 1+5.30e3T+6.93e7T2 1 + 5.30e3T + 6.93e7T^{2}
41 1+3.30e3T+1.15e8T2 1 + 3.30e3T + 1.15e8T^{2}
43 1−1.95e4T+1.47e8T2 1 - 1.95e4T + 1.47e8T^{2}
47 1+1.54e4T+2.29e8T2 1 + 1.54e4T + 2.29e8T^{2}
53 1+1.73e4T+4.18e8T2 1 + 1.73e4T + 4.18e8T^{2}
59 1+5.15e4T+7.14e8T2 1 + 5.15e4T + 7.14e8T^{2}
61 1+3.00e4T+8.44e8T2 1 + 3.00e4T + 8.44e8T^{2}
67 1−3.35e4T+1.35e9T2 1 - 3.35e4T + 1.35e9T^{2}
71 1+7.41e4T+1.80e9T2 1 + 7.41e4T + 1.80e9T^{2}
73 1+3.94e4T+2.07e9T2 1 + 3.94e4T + 2.07e9T^{2}
79 1+2.63e4T+3.07e9T2 1 + 2.63e4T + 3.07e9T^{2}
83 1+5.54e4T+3.93e9T2 1 + 5.54e4T + 3.93e9T^{2}
89 1+1.19e4T+5.58e9T2 1 + 1.19e4T + 5.58e9T^{2}
97 1+1.23e5T+8.58e9T2 1 + 1.23e5T + 8.58e9T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.970779690624769721229100637775, −8.370260135548802570355111721126, −7.65671564902208825555995612861, −6.96600054434470613955627628373, −5.89364318032810331877057677762, −4.46308109862332177505595571408, −3.06080289600461421282673557320, −1.87827805639411217032559007564, −1.36082101776373264380946692242, 0, 1.36082101776373264380946692242, 1.87827805639411217032559007564, 3.06080289600461421282673557320, 4.46308109862332177505595571408, 5.89364318032810331877057677762, 6.96600054434470613955627628373, 7.65671564902208825555995612861, 8.370260135548802570355111721126, 8.970779690624769721229100637775

Graph of the ZZ-function along the critical line